Quadri-Figures in Cayley-Klein Planes: All Around the Newton Line

TL;DR

This paper explores the extension of classical Euclidean theorems related to the Newton line and tetragons to elliptic and hyperbolic metric planes, analyzing their geometric properties beyond Euclidean space.

Contribution

It investigates the transferability of Euclidean tetragon theorems to non-Euclidean geometries, broadening understanding of geometric configurations in Cayley-Klein planes.

Findings

Classical Euclidean tetragon theorems are partially transferable to elliptic and hyperbolic planes.

Identification of conditions under which Newton line properties hold in non-Euclidean geometries.

Extension of nine-point conic theorem to Cayley-Klein planes.

Abstract

The Newton line and the associated theorems by Newton and Gauss for tetragons and quadrilaterals are closely linked to some other theorems of Euclidean geometry: a theorem by Bocher on the existence of a nine-point conic of a quadrangle, a theorem by Shatunov and Tokarev, and a theorem by Anne. This paper examines to which extent all these theorems can be transferred to other metric planes, in particular the elliptic and hyperbolic planes.